Speaker: Angelica Cueto, Ohio State University
Title: Tropical geometry of genus 2 curves
Abstract: In this talk, I will discuss the structure of tropical and non-Archimedean analytic genus 2 curves and their moduli from three perspectives: as 2-to-1 covers of $P^1$ branched at 6 points, as solutions to the hyperelliptic equation $y^2=f(x)$ where $f$ has degree 5 and as metric graphs dual to genus 2 nodal algebraic curves over a valued field. Our first description allows us to give an explicit combinatorial rule to characterize each such metric graph together with a harmonic map to a metric tree on 6 leaves, in terms of the valuations of 6 branch points in $P^1.$ Even though the tropicalization of a plane hyperelliptic curve shows no genus, we provide explicit re-embeddings of the input planar hyperelliptic curve that reveals the correct metric graphs.
From our third viewpoint, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates on the tropical moduli space, we propose an alternative set of invariants that provides new length data. This is joint work with Hannah Markwig.